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1. Heart rate during laughter. Laughter is often called “the best medicine,” since studies have
shown that laughter can reduce muscle tension and increase oxygenation of the blood. In the
International Journal of Obesity (January 2007), researchers at Vanderbilt University investigated
the physiological changes that accompany laughter. Ninety subjects (18 – 34 years old) watched
film clips designed to evoke laughter. During the laughing period, the researchers measured the

heart rate (beats per minute) of each subject with the following summary results: �̅� = 73.5, s = 6.
(NOTE: �̅� and s denote the mean and standard deviation, respectively). It is well known that the
mean resting heart rate of adults is 71 beats/minute. At ∝ = 0.05, is there sufficient evidence to
indicate that the true mean heart rate during laughter exceeds 71 beats/minute?

a. State the null hypothesis (H0): _____________________________

b. State the alternative hypothesis (HA): _____________________________

c. Calculate the test statistics (z): __________________________________

d. Write decision rule: ___________________________________________

e. State the decision (Reject or do not reject H0): _____________________

Last Name: _______________________________________ First Name: _________________________________

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2. Perform least squares estimation using the data from the table below, and write your answers as
requested (attach your calculations on separate pages):

X y

0 1

3 7

5 12

Write your answers in the blank provided and show your manual calculations on separate pages (if
necessary).

a. Regression equation: �̂� = _________________________________

Complete the following analysis of variance (ANOVA) table (show calculations on a separate
page):

Source Degrees of
Freedom (d.f.)

Sum of Squares
(SS)

Mean Square
(MS)

F

Model

Error

Total

b. Root MSE: ________

c. R-Square: ________

i. Interpretation: __________________________________________

__________________________________________

d. R: ________

i. Interpretation: __________________________________________
__________________________________________

e. Estimate the value for y when x = 3.75: ___________________________

Last Name: _______________________________________ First Name: _________________________________

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3. Earnings of Mexican street vendors: Detailed interviews were conducted with over 1,000 street
vendors in the city of Puebla, Mexico, in order to study the factors influencing vendors’ incomes
(World Development, February 1998). Vendors were defined as individuals working in the street,
and included vendors with cars and stands on wheels and excluded beggars, drug dealers, and
prostitutes. The research collected data on gender, age, hours worked per day, annual earnings,
and education level. A subset of these appear in the table below.

VenNum Earnings Age Hours

21 2841 29 12

53 1876 21 8

60 2934 62 10

184 1552 18 10

263 3065 40 11

281 3670 50 11

354 2005 65

5

401 3215 44 8

515 1930 17 8

633 2010 70

6

677 3111 20 9

710 2882 29 9

800 1683 15 5

914 1817 14 7

997 4066 33 12

a. Write a first-order model for mean annual earnings, E(y), as a function of age (x1) and
hours worked (x2).

_________

___________________________________________________________

Answer the questions below using data from the printout shown on the final page.

b. Find the least squares prediction equation: _________________________________

Last Name: _______________________________________ First Name: _________________________________

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c. Interpret the estimated β coefficients in your model:

i. Β0: ___________________________________________________________

___________________________________________________________

ii. Β1: ___________________________________________________________

d. Conduct a test of the global utility of the model (at ∝ = 0.01). Interpret the result.

___________________________________________________________
___________________________________________________________

e. Find 𝑟𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑
2 : ______________

Interpretation: _______________________________________________________

_______________________________________________________

f. Find s (estimated standard deviation of the error term): ______________

Interpretation: _______________________________________________________
_______________________________________________________

g. Is age (x1) a statistically useful predictor of annual earnings? Test using ∝ = 0.01.

Yes or No: ____________

Last Name: _______________________________________ First Name: _________________________________

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SUMMARY OUTPUT

Regression Statistics

Multiple R 0.763

R Square 0.582

Adjusted R Square 0.513

Standard Error 547.74

Observations 15

ANOVA

df SS MS F Significance F

Regression 2 5,018,231.54 2,509,115.77 8.36 0.0053

Residual 12 3,600,196.19 300,016.35

Total 14 8,618,427.73

Coefficient
s Standard Error t Stat

P-
value Lower 95%

Upper
95%

Intercept (20.35) 652.75 -0.03 0.98 -1442.56 1,401.86

Age 13.35 7.67 1.74 0.11 -3.36 30.07

Hours 243.71 63.51 3.84 0.00 105.33 382.09

Last Name: _______________________________________ First Name: _________________________________

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4. Several values are missing from the table below. Find values for the following missing values
denoted by letters a through k:

a. _____

b. _____

c. _____

d. _____

e. _____

f. _____

g. _____

h. _____

i. _____

j. _____

k. _____

SUMMARY OUTPUT
Regression Statistics

R A

R Square B

Adjusted R Square 0.949

Standard Error 88.915

Observations C

ANOVA
df SS MS F Significance F

Regression 3 4,578,427.37 f h 0.0000

Residual D e g

Total 31 4,799,789.50

Coefficients Standard Error t Stat P-value

Intercept 320.46 295.14 1.09 0.2868

AGE i 2.03 0.43 0.6690

NUMBIDS -93.26 j -3.12 0.0042

AGE-BID 1.30 0.21 K 0.0000